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Domination Defect in Graphs: Guarding With Fewer Guards

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Abstract

In this paper, we introduce a new graph parameter called the domination defect of a graph. The domination number γ of a graph G is the minimum number of vertices required to dominate the vertices of G. Due to the minimality of γ, if a set of vertices of G has cardinality less than γ then there are vertices of G that are not dominated by that set. The k-domination defect of G is the minimum number of vertices which are left un-dominated by a subset of γ - k vertices of G. We study different bounds on the k-domination defect of a graph G with respect to the domination number, order, degree sequence, graph homomorphisms and the existence of efficient dominating sets. We also characterize the graphs whose domination defect is 1 and find exact values of the domination defect for some particular classes of graphs.

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Authors and Affiliations

  1. Department of Mathematics, St. Xavier’s College, Kolkata, India

    Angsuman Das

  2. Department of Mathematics, University of Johannesburg, Auckland Park, South Africa

    Wyatt J. Desormeaux

Authors
  1. Angsuman Das
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  2. Wyatt J. Desormeaux
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Corresponding author

Correspondence to Angsuman Das.

Additional information

The research of the first author is partially supported by NBHM Research Project Grant, (Sanction No. 2/48(10)/2013/ NBHM(R.P.)/R&D II/695), Govt. of India.

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Das, A., Desormeaux, W.J. Domination Defect in Graphs: Guarding With Fewer Guards. Indian J Pure Appl Math 49, 349–364 (2018). https://doi.org/10.1007/s13226-018-0273-8

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  • DOI: https://doi.org/10.1007/s13226-018-0273-8

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